The following is the main result in the paper by Vinnikov, Putinar, Alpay: A Hilbert space approach to bounded analytic extension in the ball, 2003, Communications on Pure and Applied Analysis. The authors proved it by using Nevanlinna-Pick interpolation theorem and several results from functional analysis.
Let $F:\mathbb{D}\rightarrow\mathbb{B}$ be an injective holomorphic map. Here $\mathbb{D}$ and $\mathbb{B}$ denote the unit disk in $\mathbb{C}$ and the unit ball in $\mathbb{C}^{2}$, respectively. Assume further that $F$ can be extended to be $C^{1}$ on an open neighbourhood $U$ of $\mathbb{D}$ such that $F\left(z\right)\in\partial\mathbb{B}$ if and only if $z\in\partial\mathbb{D}$; and moreover $$ \left\langle F'\left(z\right),F\left(z\right)\right\rangle \neq0,\forall z\in\partial\mathbb{D}. $$ Prove that $$ \left\langle F\left(u\right),F\left(v\right)\right\rangle =\left\langle u,v\right\rangle ;\forall u,v\in\mathbb{D}. $$ Here $\left\langle \cdot,\cdot\right\rangle $ is the standard inner product, $\left\langle z,w\right\rangle =z_{1}\overline{w_{1}}+\ldots+z_{n}\overline{w_{n}}$ if $z,w\in\mathbb{C}^{n};$ and $\partial U$ denotes the boundary of $U.$
Do anyone have another simple approach for this problem?
